Problems in Quantum Mechanics: Third Edition
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Suitable for advanced undergraduates and graduate students of physics, this third edition was edited by Dirk ter Haar, a Fellow of Magdalen College and Reader in Theoretical Physics at the University of Oxford. This enlarged and revised edition includes additional problems from Oxford University Examination papers. The book can be used either in conjunction with another text or as advanced reading for anyone familiar with the basic ideas of quantum mechanics. 1975 edition.
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Problems in Quantum Mechanics - Dover Publications
PROBLEMS IN
QUANTUM
MECHANICS
THIRD EDITION
Edited by
D. ter Haar
Dover Publications, Inc.
Mineola, New York
Copyright
Copyright © 1975 by Pion Ltd.
All rights reserved.
Bibliographical Note
This Dover edition, first published in 2014, is an unabridged republication of the third revised and enlarged edition of the work, originally published by Pion Ltd., London, in 1975.
This Dover edition is published by special arrangement with Pion Ltd., 207 Brondesbury Park, London NW2 5JN, England.
International Standard Book Number
eISBN-13: 978-0-486-79505-8
Manufactured in the United States by Courier Corporation
78080501 2014
www.doverpublications.com
Contents
Preface
1One dimensional motion
2Tunnel effect
3Commutation relations; Heisenberg relations; spreading of wave packets; operators
4Angular momentum; spin
5Central field of force
6Motion of particles in a magnetic field
7Atoms
8Molecules
9Scattering
10Creation and annihilation operators; density matrix
11Relativistic wave equations
Subject index
Preface to second edition
This is essentially an enlarged and revised second edition of a collection of problems which consisted of a text by Gol’d man and Krivchenkov augmented by a selection from a similar text by Kogan and Galitskii.
In preparing the present edition I have used the opportunity to revise some of the problems in the first edition, to change a few of the solutions, and to make the notation both uniform and conforming to English usage. Also, I have added a few problems from a collection by Irodov on atomic physics and a number of new problems which were mainly taken from Oxford University Examination papers. I should like to express my thanks to the Oxford University Press for permission to include these problems.
These problems can be used either in conjunction with any modern textbook, such as those by Schiff, Kramers, Landau and Lifshitz, Messiah, or Davydov, or as advanced reading for anybody who is familiar with the basic ideas of quantum mechanics from a more elementary textbook.
Oxford,
September 1963
D. ter Haar
Preface to third edition
In preparing the third edition I have dropped some of the problems, slightly rearranged the order of the problems, and added new problems to the old chapters, as well as added new sections on the density matrix and annihilation and creation operator problems and on relativistic wave equations. Otherwise the aim and scope of the book remain much as they were before, but to help readers I have added stars to more complicated problems.
Oxford,
September 1974
D. ter Haar
problems
1
One-dimensional motion
1. Determine the energy levels and the normalised wave functions of a particle in a potential weir
. The potential energy V of the particle is:
2. Show that for particles in a potential well
(see preceding problem) the following relations hold:
Show also that for large values of n the above result agrees with the corresponding classical result.
3. Determine the momentum probability distribution function for particles in the nth energy state in a potential well
.
4. A particle in an infinitely deep rectangular potential well is in a state described by the wave function
where a is the well width and A a constant.
Find the probability distribution for the different energies of the particle and also the average value and the dispersion of the energy.
5. A particle is in the ground state in a potential well of length a. At time t = 0 the wall at x = a is suddenly moved to x = 2a. Calculate the probability that, at time t > 0,
(a) the energy of the particle is the same as before t = 0; and
(b) the energy of the particle is less than before t = 0.
6*. A particle is enclosed in a one-dimensional rectangular potential well with infinitely high walls. Evaluate the average force exerted by the particle on the wall of the well.
7. Determine the energy levels and wave functions of a particle in an asymmetrical potential well (see fig. 1). Consider the case where V1=V2.
Fig. 1.
In order to eliminate ħ, μ, and ω ,†
and the energy E will be expressed in units ħω (Eħω). The Schrödinger equation for the oscillator in the new variables will be of the form
to show that
(b) Determine the normalised wave functions and the energy levels of the oscillator.
. Express the wave function of the nth excited state in terms of the wave function of the ground state using the operator â.
in the energy representation.
Hint.
9. Using the results of the preceding problem, show by direct multiplication of matrices that for an oscillator in the nth stationary state we have
Show that, for any stationary state, the root-mean-square value of x is the same as it would have been for a classical oscillator with the same energy.
. Determine the probability w to find the particle outside the classical limits, when it is in its ground state.
11. Find the energy levels of a particle moving in a potential of the following form:
12. Write down the Schrödinger equation for an oscillator in the "p-representation" and determine the momentum probability distribution function.
13. Find the wave functions and energy levels of a particle in a potential V(x) = V0(a/x— x/a)² (x>0) (see fig. 2) and show that the energy spectrum is the same as the oscillator spectrum.
Fig. 2.
14*. Determine the energy levels for a particle in a potential V = –V0/cosh²(x/a) (see fig. 3).
Fig. 3.
15*. Determine the energy levels and wave functions for a particle in the potential V = V0cot²(πx/a) (0<x<a) (see fig. 4), and derive the normalisation constant of the ground state wave function.
Consider the limiting cases of small and large values of V0.
Fig. 4.
16. Determine the energy levels for a particle in a potential
17. An electron of mass μ moves in a one-dimensional potential
where P is a positive dimensionless constant, δ(x) the Dirac delta-function, and a a constant length. Discuss the bound states for this potential as a function of P.
18*. A one-dimensional hydrogen
atom is one in which an electron confined to the x-axis is acted upon by a force inversely proportional to the square of its distance from the origin. Find the energy eigenvalues and the eigenfunctions of this system.
19. Determine the wave functions of a charged particle in a uniform field V(x) = –Fx.
20*. Find the wave functions and energy levels of the stationary states of a particle of mass μ in a uniform gravitational field g for the case when the region of the motion of the particle is limited from below by a perfectly reflecting plane. (As a classical analogy of this system we can take a heavy solid ball, bobbing up and down on a metallic plate. We note that all calculations and results of this problem are clearly correct also for the case of the motion of a particle of charge e , in the presence of a reflecting plane, provided we replace in all equations g by (e/m) Take the limit to classical mechanics.
21*. Derive expressions for the wavefunction of a particle moving in a potential V(x) using the semi-classical approximation. Give conditions for the applicability of the approximation and determine the quantization condition.
22. Use the semi-classical approximation to derive an expression for the number of discrete levels of a particle moving in a given potential.
23. Determine in the semi-classical approximation the energy spectrum of a particle in the following potentials:
(a(oscillator);
(b) V = V0cot²(πx/a) (0 < x < a).
24. Use the semi-classical approximation to determine the bound energy levels for a particle of mass μ moving in a potential which equals –V0 for x = 0, changes linearly with x until it vanishes at x = ±a, and is zero for |x > a (see fig. 5).| Determine also the total number of discrete energy levels.
Fig. 5.
25. Determine in the semi-classical approximation the average value of the kinetic energy in a stationary state.
26. Use the result of the preceding question to find in the semi-classical approximation the average kinetic energy of a particle in the following potentials:
(a;
(b) V = V0cot²(πx/a) (0 < x < a) (see problem 23).
27. Determine the form of the energy spectrum of a particle in a potential V(x) = axv, using the semi-classical approximation and applying the virial theorem.
28. Obtain the semi-classical expression for the energy levels of a particle in a uniform gravitational field for the case where its motion is limited from below by a perfectly reflecting plane.
29. A particle oscillates in a one-dimensional potential field between two turning points x = a and x = b. The former is due to a vertical potential wall, while the latter is of the more usual type with dV/dx finite. Apply the WKB method to find the quantisation condition for a stationary state in such a potential.
30*. Determine in the semi-classical approximation the form of the potential energy V(x) for a given energy spectrum En. V(x) may be assumed to be an even function V(x)=V (–x), which increases monotonically for x > 0.
31*. Find the semi-classical solution of the Schrödinger equation in the momentum-representation.
Show that the same semi-classical function is obtained by going over from the "x-representation to the
p-representation" starting from the usual semi-classical coordinate wave function.
32. Find the wave functions and energy levels of the stationary states of a plane rotator with moment of inertia I.
A rotator is a system of two rigidly connected particles rotating in a plane (or in space). The moment of inertia of a rotator is equal to I=μa², where μ is the reduced mass of the particles and a their distance apart.
33. If the wave function of a plane rotator at t = 0 is given by ψ(φ,0) = A sin² φ, where A is a normalising constant, what will be the wave function ψ (φ, t) at time t ?
34. A bead of mass μ is confined to a thin wire which forms a rigid circular loop of radius a. Find an expression for the tension in the wire when the system is in a stationary state, assuming the wire to be unstressed before the bead is placed on it.
35. Write down the Schrödinger equation in the "p-representation" for a particle moving in a periodic potential V(x) = V0cos bx.
36. Write down the Schrödinger equation in the "p-representation" for a particle moving in a periodic potential V(x) = V(x + b).
37*. Determine the allowed energy bands of a particle moving in the periodic potential given by fig. 6. Investigate the limiting case where V0→∞, and b→0 while V0b = constant.
Fig. 6.
38*. A simple model of the electronic energy levels in a metal uses a one-dimensional potential of the form
where μ is the electron mass, a the lattice constant, P a positive, dimensionless constant, and δ(x) the Dirac delta-function. Find expressions for the effective mass at the upper band edges.
39*. A particle moves in a periodic field V(x):
Using a suitable semi-classical approximation obtain a transcendental equation to determine the allowed energy bands. Discuss this equation.
40. Show that for particles scattered by a complex potential, V(x)(1 + iξ), the probability current density,
and the probability density, ρ= ψ * ψ, satisfy the continuity
equation
41. Use a variational principle to prove that any purely attractive one-dimensional potential has at least one bound state.
42. A particle of mass μ moves in a one-dimensional potential λV(x), where V(x) satisfies the conditions
Prove that, if λ is sufficiently small, there exists a bound state with an energy E which is approximately given by
.
2
Tunnel effect
1. In studying the emission of electrons from metals, it is necessary to take into account the fact that electrons with an energy sufficient to leave the metal may be reflected at the metal surface. Consider a one-dimensional model with a potential V which is equal to –V0 for x < 0 (inside the metal) and equal to zero for x > 0 (outside the metal) (see fig. 7), and determine the reflection coefficient at the metal surface for an electron with energy E > 0.
Fig. 7.
2*. In the preceding problem it was assumed that the potential changed discontinuously at the metal surface. In a real metal this change in potential takes place continuously over a region of the dimensions of the order of the interatomic distance in the metal. Approximate the potential near the metal surface by the function
Fig. 8.
and determine the reflection coefficient of an electron with energy E > 0.
3. Determine the coefficient of transmission of a particle through a rectangular barrier (see fig. 9).
Fig. 9.
4. Determine the coefficient of reflection of a particle by a rectangular barrier in the case where E > V0 (reflection above the barrier).
5. A particle is moving along the x-axis. Find the probability for transmission of the particle through a delta-function potential barrier at the origin.
Fig. 10.
7*. Find the coefficient of transmission of a particle through a triangular barrier (see fig. 11). Consider the limiting cases of small and of large pentrability.
Fig. 11.
8*. Calculate the coefficient of transmission through a potential barrier V(x) = V0/cosh² (x/a) (see fig. 12) for particles moving with an energy E0.
Fig. 12.
9. Evaluate in the semi-classical approximation the transmission coefficient for a parabolic potential barrier of the following form (see fig. 13):
Give the criterion for the applicability of the result obtained.
Fig. 13.
10. Calculate in the semi-classical approximation the coefficient of transmission of electrons through a metal surface under the action of a large electrical field strength F (fig. 14). Find the limits of applicability of the calculation.
Fig. 14.
11*. The change of the potential near a metal surface is in reality a continuous one. For instance, the electrical image potential Ve im = – e/4x will act at large distances from the surface. Determine the coefficient of transmission D of electrons through a metal surface under the action of an electrical field, taking into account the electrical image force (fig. 15).
Fig. 15.
12*. A symmetrical potential V(x) consists of two potential wells separated by a barrier (see fig. 16). Assuming that one may use a semi-classical argument, determine the energy levels of a particle in the potential V(x). Compare the energy spectrum obtained with the energy spectrum of a single well.
Fig. 16.
13. Assume that at t = 0 there exists an impenetrable partition between the two symmetrical potential wells (see preceding problem) and that a particle is in a stationary state in the well on the left.
Determine the time τ it takes after the partition is removed before the particle will be in the well on the right.
14*. The potential V(x) consists of N identical potential wells separated by identical potential barriers (see fig. 17). Determine the energy levels in this potential, assuming that one can use the semi-classical approach.
Compare the energy spectrum obtained with the energy spectrum of a single well.
Fig. 17.
15*. Assuming that one may use the semi-classical approach, find the quasi-stationary levels of a particle in the symmetrical field given by fig. 18.
Fig. 18.
Find also the transmission coefficient D(E) for a particle with energy E0, where V0 is the maximum value of the potential V(x).
16. (i) Show generally that for any barrier the relation R+D = 1 is automatically satisfied, where R is the reflection coefficient and D the transmission coefficient.
(ii) A particle confined to one dimension encounters a symmetric potential barrier of finite extension with its centre at x = 0. For each energy E = ħ²k²/2μ there is an even and an odd solution of the Schrödinger equation which to the right of the barrier have the form A cos(kx + φ) and B sin(kx+ φ'), respectively. Show that the reflection and transmission coefficients at this energy are, respectively, sin²2(φ – φ') and cos²2(φ – φ').
17. A particle is moving in a potential V(x) which is zero for |x| > a and finite for |x| < a. Check that the wavefunction ψ(x) satisfies the integral equation
Obtain an expression for the reflection coefficient to lowest order in V, if a beam of particles of mass μ and momentum p = ħk is incident upon this potential V(x).
3
Commutation relations; Heisenberg relations; spreading of wave packets; operators
1. Let Â, , and Ĉ with Ĉ in terms of the commutators [Â,Ĉ]
2. Show that for algebraic manipulations with commutators the distributive law holds, that is, that the commutator of a sum is the sum of commutators:
3. Prove that if f(x) is a function of the coordinate x and g(p) a function of the momentum p, we have
4. Prove the following relation:
where K is a c number and if λ is a c-number,
6. Prove that
this relation is called Kubo’s identity.
7. Show that if the two Hermitean operators  the following relation will hold:
and F
Hint. Express F) in a Taylor series.
9. Use the Heisenberg relations to estimate the ground state energy of a harmonic oscillator.
10. Estimate the energy of an electron in the K-shell of an atom of atomic number Z both for the relativistic and the non-relativistic case.
11. Estimate the ground state energy of a two-electron atom with nuclear charge Z, using the Heisenberg relations.
12. The magnetic field produced by a free electron is partly due to its motion and partly due to the presence of its intrinsic magnetic moment.
It is known from electrodynamics that the magnetic field strength H1 due to a moving charge is of the order of magnitude
and the magnetic field strength H2 due to a dipole moment μ
To determine the magnetic moment μ of a free electron from a measurement of the field strength produced by it, it is necessary that the following two conditions are satisfied:
and
The meaning of the last condition is that the electron must be localised in a region ∆r which is much smaller than the distance from that region to the point where the magnetic field is observed.
Is it possible to satisfy these two conditions simultaneously ?
Hint. Take into account the Heisenberg relations and the value of the electronic magnetic moment μ = eħ/2mc.
13. We consider a particle in a one-dimensional symmetrical potential well in which there is always, as is well known, at least one energy level (compare problem 41 of section 1).
If for a given depth of the well V0 its width a is reduced until it satisfies the inequality
then, at first sight, the spatial localisation of a particle bound in the well will become much more precise (∆x ~ a)> and as the spread in momentum ∆p in any case is limited to a value of the order √(μV0), the following
inequality
would hold, violating the Heisenberg relations.
Show the error in the above argument and evaluate the product of the spread in coordinate and the spread in momentum of the particle.
14. The trajectory of a particle in a Wilson chamber is a chain of small droplets of linear dimensions of about 1 μ. Could one observe deviations from the classical motion for an electron with energy 1 keV ?
15. For what value of the relative uncertainty in the angular momentum of the electron in the first Bohr orbit does its angular coordinate become completely undetermined ?
(compare the derivation of the Heisenberg relations in problem 7 of section 3).
17. The wave function of a free particle at t = 0 is given by the expression
The function φ(x) is real and appreciably different from zero only for values of x within the interval —δ+ δ. Determine in which region of x-values the wave function will be different from zero at time t.
18. Find the change of a wave function which is given at t = 0 for the following three cases (spreading of a wave packet):
(a) a free particle,
*(b) a particle moving in a uniform field,
19. Show that the problem of how to determine the motion of an oscillator under the action of an external force f(t) can be reduced to the simpler problem of determining the motion of a free oscillator, if we introduce the new variable x1=x – ξ(t), where ξ(t
20*. Find the Green function for an oscillator whose eigenfrequency changes with time, and express it in terms of the solution of the classical equation of motion of an oscillator of varying frequency.
21*. Use the Green function obtained in the preceding problem to determine the time dependence of the probability density for particles moving in a potential
The particle wave function at t = 0 is equal to
22*. Find the Green function of an oscillator whose eigenfrequency changes in time and which is acted upon by a perturbing force f(t).
23*. An oscillator is at t = 0 in its nth energy eigenstate. Determine the probability that it will make a transition to the mth eigenstate under the action of a perturbing force f(t). Find the average value and the dispersion of its energy at time t.
24*. The point of suspension x0 of an oscillator in its ground state starts to move at t = 0. The point of suspension moves according to the law x0 = x0(t). At t = T the point of suspension is fixed again.
Use the Green function (see problem 20 of section 3) to find the wave function of the system at any time t > 0, and also the probability of excitation of the nth level as a result of the process considered.
where ω is the frequency of the oscillator).
25*. Discuss the motion of a particle governed by the Hamiltonian
where γ is a positive constant.
Find the wave function for this system for the case where F(t) = 0 as t→∞, if at t = 0 the wavefunction is given by the formula
26. Inasmuch as the Schrödinger equation is a first-order differential equation with respect to time, ψ(t) is uniquely determined by the value of ψ(0). Write this connection in the form
where Ŝ(t) is some operator.
(a) Show that the operator Ŝ(t) satisfies the equation
and is a unitary operator, that is, Ŝ+ = Ŝ–1.
(b) Show that in the case where Ĥ does not depend on time, Ŝ(t) is of the form
27. at time t follows from the following expression:
(a) Show that the time dependence of the operatorwith Ŝ(t) determined by the equation Ŝ(t)ψ(0) = ψ(t), satisfies the equation
(b) Prove the following operator equation:
where
satisfy the commutation relation
the corresponding time-dependent operators satisfy the equation
28. Derive a law for the differentiation with respect to the time of the product of two operators.
29. Find the coordinate operator for the case of a free particle in the Heisenberg representation.
30. Find the coordinate and momentum operators in the Heisenberg representation for the case of a linear harmonic oscillator by solving the equations of motion for these operators.
31. In the Heisenberg picture the operators are time-dependent and satisfy the equation of motion (see problem 27 of section 3)
while the wavefunction is time-independent. On the other hand, in the Schrödinger picture the operators (unless explicitly dependent on the time) are time-independent, while the wavefunction satisfies the
Schrödinger equation
The operators and wavefunctions in the Heisenberg picture (indicated by a subscript H) are related to those in the Schrödinger picture (indicated by the subscript S) through the formulae
It is often convenient to work in an intermediate picture, called the interaction picture. This is the case when the Hamiltonian is of the form
In that case we can introduce operators and wavefunctions in the interaction picture (indicated by a subscript int) as follows
and ψint.
32. Prove for the one-dimensional case that if q and p are the generalized coordinate and momentum of a particle, we have
where H is the Hamiltonian of the system, while the bar indicates a quantum-mechanical average. This is a particular example of the Ehrenfest theorem, which states that the classical equations of motion still hold, if we replace the physical quantities by quantum-mechanical averages.
33. Show that the average value of the time derivative of a physical quantity, which does not explicitly depend on the time, is equal to zero in a stationary state of the discrete spectrum.
34. Prove the virial theorem in quantum mechanics.
35. What is the physical meaning of the quantity p0 in the expression for the wave function
if φ(x) is a real function ?
36. Show that the average value of the momentum in a stationary state with a discrete energy eigenvalue is equal to zero.
37. Determine the time dependence of the coordinate